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LECTURE TEXT
First-order differential equations in chemistry
Gudrun Scholz • Fritz Scholz
Received: 7 August 2014 / Accepted: 13 September 2014
� Springer International Publishing 2014
Abstract Many processes and phenomena in chemistry,
and generally in sciences, can be described by first-order
differential equations. These equations are the most
important and most frequently used to describe natural
laws. Although the math is the same in all cases, the stu-
dent may not always easily realize the similarities because
the relevant equations appear in different topics and con-
tain different quantities and units. This text was written to
present a unified view on various examples; all of them can
be mathematically described by first-order differential
equations. The following examples are discussed: the
Bouguer–Lambert–Beer law in spectroscopy, time con-
stants of sensors, chemical reaction kinetics, radioactive
decay, relaxation in nuclear magnetic resonance, and the
RC constant of an electrode.
Keywords Differential equations � Bouguer–Lambert–
Beer law � Time constants � Chemical kinetics �Radioactive decay � Nuclear magnetic resonance � RC
constant
Introduction
‘‘Differential equations are extremely important in
the history of mathematics and science, because the
laws of nature are generally expressed in terms of
differential equations. Differential equations are the
means by which scientists describe and understand
the world’’ [1].
The mathematical description of various processes in
chemistry and physics is possible by describing them with
the help of differential equations which are based on simple
model assumptions and defining the boundary conditions
[2, 3]. In many cases, first-order differential equations are
completely describing the variation dy of a function y(x)
and other quantities. If y is a quantity depending on x, a
model may be based on the following assumptions: The
differential decrease of the variable y is proportional to a
differential increase of the other variable, here x, i.e.
-dy * dx. This decrease -dy should depend on the
function y itself: -dy * ydx, and together with a so far
unknown constant a, results in the equation
dy ¼ �aydx ð1Þ
Thus follows the ordinary linear homogeneous first-
order differential equation:
dy
dxþ ay ¼ 0 ð2Þ
The characteristics of an ordinary linear homogeneous
first-order differential equation are: (i) there is only one
independent variable, i.e. here x, rendering it an ordinary
differential equation, (ii) the depending variable, i.e. here y,
having the exponent 1, rendering it a linear differential
equation, and (iii) there are only terms containing the
Electronic supplementary material The online version of thisarticle (doi:10.1007/s40828-014-0001-x) contains supplementarymaterial, which is available to authorized users.
G. Scholz
Department of Chemistry, Humboldt-Universitat zu Berlin,
Brook-Taylor-Str. 2, 12489 Berlin, Germany
F. Scholz (&)
Institute of Biochemistry, University of Greifswald,
Felix-Hausdorff-Str. 4, 17487 Greifswald, Germany
e-mail: [email protected]
123
ChemTexts (2014) 1:1
DOI 10.1007/s40828-014-0001-x
variable y and its first derivative, rendering it a homoge-
neous first-order differential equation.
This equation can be solved when, e.g. the boundary
conditions are such that y varies between y0 and y, when x
varies between 0 and x. Following a separation of vari-
ables, the integration of Eq. 2 gives:
dy
y¼ �adx ð3Þ
Zy
y0
dy
y¼ �a
Zx
0
dx ð4Þ
lny
y0
¼ �ax ð5Þ
y ¼ y0e�ax ð6Þ
Equation 6 describes the exponential decrease of y as a
function of x.
This formalism will now be applied to some special
cases which occur frequently in chemistry, and finally all
discussed cases will be compared in a table.
The Bouguer–Lambert–Beer Law
The intensity of electromagnetic radiation (e.g. visible
light), i.e. exactly the radiant flux I (unit W, watt or J s-1,
joule per second), diminishes along the path length
x through a homogeneous absorbing medium (e.g. a col-
oured solution). Figure 1 depicts a cuvette and the changes
in radiant flux along the optical path length.
The differential decrease -dI of radiant flux by passing
through the differential length increment dx is supposed to
be proportional to the actual value of I at xi. To understand
this, we must consider the physical background of the
decrease of radiant flux: If the radiation is understood as a
flux of photons, the absorption of radiation is the loss of
photons due to their ‘‘capture’’ by absorbing particles
(molecules, atoms or ions) in the cuvette. Clearly, the
effectivity of capture must be proportional to the number of
particles per volume, i.e. their concentration c in mol L-1,
as the probability that a photon hits a particle will be
proportional to its concentration. However, not each hitting
leads to an absorption event (capture of a photon). To take
into account the probability that a collision of a photon
with a particle leads to its capture, one defines an effective
cross section of the particles. This cross section has the unit
of an area because one may understand it as an effective
target area for the photons in contrast to the geometric
target area which a particle exposes to the photon flux.
Instead of using the effective cross section, one may define
a constant j (Greek letter kappa) which theoretically can
have values between 0 and 1, giving the fraction of suc-
cessful absorption events. j is a value specific for the
particles and specific for the photon energy Ephoton, and
thus the frequency m (Greek letter nu) of the radiation, with
m = Ephoton/h (h is the Planck constant
6.62606957(29) 9 10-34 Js), and the wavelength k (Greek
letter lambda) with k ¼ hclight
�Ephoton (clight being the
velocity of light in the respective medium).
From the preceding discussion follows that the differ-
ential equation
�dIðxÞ ¼ IðxÞjcdx ð7Þ
adequately describes the decrease of radiation flux. Since jis specific for the energy of absorbed photons, this equation
Fig. 1 Electromagnetic radiation is trespassing a cuvette filled with a
homogeneous absorbing medium. I0 is the radiant flux before entering
the cuvette, I is the radiant flux leaving the cuvette, �dI is the
differential decrease of radiant flux by passing through the differential
length increment dx at xi
Fig. 2 The decay of radiation flux when passing through the
absorbing medium
1 Page 2 of 12 ChemTexts (2014) 1:1
123
relates to monochromatic radiation (radiation with one
constant frequency, i.e. photon energy). The meaning of
Eq. 7 can be understood with the help of Fig. 2: If xe marks
the overall length which the electromagnetic radiation
passes through the absorbing medium, and the intensity
(radiation flux) of the incident light is I0 (at x ¼ 0), then at
a path length xi the intensity of light will have dropped to Ii
and the slope of IðxÞ ¼ f ðxÞ, i.e. dIðxÞdx
will be proportional
to Ii and j and c.
Integration of Eq. 7 and some rearrangements have to be
performed as follows:
� dIðxÞIðxÞ ¼ jcdx ð8Þ
�ZI
I0
dIðxÞIðxÞ ¼ jc
Zxe
0
dx ð9Þ
� lnI
I0
¼ jcxe ð10Þ
lnI0
I¼ jcxe ð11Þ
logI0
I¼ 1
ln 10jcxe � 0:4343jcxe ð12Þ
The ratio log I0
Iis called absorbance A, and the product
0:4343j is called the molar absorption coefficient e (Greek
letter epsilon) or molar absorptivity. The path length of the
radiation xe is usually given the symbol l. The Bouguer–
Lambert–Beer Law is thus normally written as:
A ¼ ecl ð13Þ
Outside of this purely mathematical analysis, it needs
to be mentioned that Eq. 13 has a restricted range of
validity: it is a good description of real systems only
at low concentrations. At higher concentrations
(sometimes already above 10-5 mol L-1) intermo-
lecular interactions of the absorbing particles, and
chemical equilibria can lead to deviations (apparent
variations of the molar absorption coefficient). Fur-
ther, another contribution to the absorption coeffi-
cient depends on the refractive index n of the
solution. Because the refractive index may signifi-
cantly vary with the concentration of the dissolved
analyte, it is not e, which is constant, but the molar
refraction and the term en=ðn2 þ 2Þ2 should be used
instead of e [4].
The time constant of a sensor
Sensors measure a physical or chemical quantity and
transduce it to an output signal which is read, monitored
or stored. Possible physical quantities are temperature,
pressure, radiative flux, magnetic field strength, etc.
Chemical quantities are mainly concentrations and activ-
ities of molecules, atoms and ions. The recorded signals
are usually voltages or currents. The most typical feature
of a signal is that the results are one dimensional, e.g. the
output signal is a single quantity, i.e. one measures only
that signal and not a dependence of that signal on another
given quantity. Most devices for chemical analysis pro-
duce two-dimensional read-outs, e.g. optical spectra in
which the absorbance is displayed as a function of
wavelength (E ¼ f ðkÞ), voltammograms in which currents
are displayed as function of electrode potential or X-ray
diffractograms, in which the intensity of diffracted rays is
displayed as function of diffraction angle, etc. In modern
instrumentation, one has even expanded the
Fig. 3 A comparison of the three common dimensionalities of analytical devices
ChemTexts (2014) 1:1 Page 3 of 12 1
123
dimensionality to three, when, as an example, optical
spectra (E ¼ f ðkÞ) (or mass spectra, i.e. ion intensities
versus the mass-to-charge ratio of ions) are displayed as a
function of elution time of a chromatogram. Figure 3
gives a comparison of the common dimensionalities of
analytical measurements.
Since any measurement needs time, there is nothing like
an instantaneous establishment of a signal. This is easy to
see when using a sensor, e.g. a pH electrode: There is
always a certain time period in which the reading changes
until we finally have the impression that a constant end
value is reached. The same is true also for two- or three-
dimensional measurements, but we cannot easily detect it
because the variation of the measured signal (e.g. the
absorbance) anyway changes as a function of the varied
quantities (e.g. the wavelength) and thus with time. Nor-
mally, the wavelength is changed with the so-called scan
rate dk=dt (rate of recording the spectrum), and generally
(see Fig. 4), the quantity x is varied with a scan rate dx=dt
(which may be also zero). Whether we measure at each
wavelength really the end value of the absorbance can be
only seen if we decrease the rate at which the wavelength is
changed (in the extreme even keeping the wavelength
constant). Referring to Fig. 4, this means in general terms,
that a variation of the scan rate dx=dt may give a repro-
ducible and identical response only below a certain limiting
rate ðdx=dtÞlimit. If that rate is exceeded, the signal cannot
establish its true value and the spectra are distorted (the
signal lags behind) (cf. Fig. 4).
Figure 4 shows impressively that it is important to know
the rate at which the signal is established for a given x
value. In case of a sensor, i.e. a one-dimensional device
where no parameters like x or y are changed, the time
change of the signal can be studied following a
concentration step. The introduction of the sensor into a
solution can be regarded as a concentration step. Figure 5
depicts two different kinds of response of a sensor on a
concentration step.
Figure 5 depicts two basic types of time responses of
sensors. The different sensor behaviours shown in B and C
can be modelled with the help of different differential
equations. Whereas the response curve shown in B can be
modelled with a first-order differential equation; the curve
shown in C needs higher-order differential equations [5]. At
this point, it is necessary to note that it is impossible to realize
a concentration step with infinite rate of concentration rise, as
shown in Fig. 5a. This means, when the temporal response
properties of a sensor are studied, this concentration rise has
to be much quicker than the response of the sensor. Further,
also the response shown in Fig. 5b is to some extend an
idealization, and in reality there may be always a sluggish
response at the start, but it may be on such short time scale
that it escapes our recognition. The response curve shown in
Fig. 5b can be modelled as follows:
S ¼ Smax � zðtÞ ð14Þ
z is a time-dependent quantity for which we write the
first-order differential equation
a1
dz
dtþ a2zðtÞ ¼ 0 ð15Þ
Integration and rearrangements follow:
Zz
z0
dz
zðtÞ ¼ �a2
a1
Z t
0
dt ð16Þ
lnzðtÞz0
¼ � a2
a1
t ð17Þ
Fig. 4 Possible distortion of a
spectrum when the rate of
changing x with time above a
limiting value ðdx=dtÞlimit
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123
zðtÞz0
¼ e�a2
a1t ð18Þ
zðtÞ ¼ z0e�a2
a1t ð19Þ
and with Eq. 14 follows:
S ¼ Smax � z0e�a2
a1t ð20Þ
Obviously, z0 should be equal to Smax, because then we
can write:
S ¼ Smax � Smaxe�a2
a1t ¼ Smax 1� e
�a2a1
t� �
ð21Þ
Since the term 1� e�a2
a1t
� �should not have a unit, it
follows that the ratio a2
a1must be a reciprocal time, and we
may write Eq. 21 using the definition a1
a2¼ s (Greek letter
tau), i.e. a2
a1¼ 1
s:
S ¼ Smax 1� e�ts
� �ð22Þ
The quantity s is called the time constant of the sensor.
At t ¼ s, the signal S has the value
S ¼ Smaxð1� e�1Þ ¼ Smax 1� 1e
� �� 0:632Smax. In other
words, after elapse of s, the signal has reached 63.2 % of
its ‘‘final value’’. Equation 22 implies of course that the
signal will never reach a constant value, but the increments
of the function z in Eq. 14 may become meaninglessly
small as the time progresses. Because of the e-function of
Eq. 22, one can give simple relations between the time to
reach 50, 63.2, 90, 99 % (and any other values) of Smax:
t63:2% ¼ s ¼ 1:44t50% ð23Þt50% ¼ 0:69s ð24Þt90% ¼ 2:3s ð25Þt99% ¼ 4:6s ð26Þ
Figure 6 shows the response curve of Fig. 5b with a
scale giving the response in percentage.
The time constant s is an important parameter of a
sensor. However, in practice, one is much more interested
in the time necessary to reach 90 or 99 % of the final value,
i.e. t90% and t99%, as the signal reached after that time is
often regarded as a good estimate of the true signal value.
Thanks to the exponential dependence of the signal on time
(Eq. 22), these time data can easily be calculated from the
time constant s according to the Eqs. 25 and 26.
In case of flow-through detectors, one can also calculate
the so-called response volume, which is simply the volume
of solution flowing through the detector within t ¼ s. It can
be calculated by vresponseðsÞ ¼ s � f , where f is the flow rate,
e.g. in ml s-1 [6]. Of course, one can also calculate the
response volumes relating to 90 or 99 % of the signal, i.e.
vresponseðsÞ ¼ t90% � f or vresponseðsÞ ¼ t99% � f , respectively.
The response volumes of a flow-through detector can be
larger or also smaller than the geometric volume of the
detector. This depends on their construction and principle
of function.
We have already mentioned that the response type
shown in Fig. 5c is observed in many experimental cases.
Fig. 5 a The concentration is stepped from zero to a constant value.
b The signal of the detector starts to respond immediately when the
step is made, and the slope of the response over time continuously
decreases until it finally approaches a constant final value. The
response curve has no turning point. c The sensor starts slowly to
respond (but with increasing rate/slope), and after a turning point the
response slows down (decreasing slope) until it approaches the final
value
Fig. 6 Time response of a sensor when it can be modelled with a
first-order differential equation
ChemTexts (2014) 1:1 Page 5 of 12 1
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For analytical applications, the most important information
to know is, how long the sensor needs to acquire, e.g. 90 or
99 % of the final value. Thus, it is much less important to
exactly describe the complete response curve from begin-
ning to the end, which would be only possible by solving
higher-order differential equations and characterizing the
response by more than one time constant. Therefore, a
frequently used approach is as follows: one analyses only
the later part of the response (i.e. the part after the turning
point) using Eq. 22, and introduces a delay time tdelay after
which this equation is assumed to be followed. The
response is then described by
S ¼ Smax 1� e�t�tdelay
s
� �ð27Þ
The response of a sensor of the type shown in Fig. 5c
and in Fig. 7 can also be described with the help of two or
more time constants using an equation like the following
(here with two time constants):
S ¼ S1 1� e� t
s1
� �þ S2 1� e
� ts2
� �ð28Þ
The crucial point is that there should be a model sup-
porting the use of two or more time constants, as otherwise
the fitting of such curve may be mathematically correct but
meaningless, as not supported by a model. A somewhat
exotic physico-chemical example where the fitting of a
response with an equation of the type of Eq. 28 is based on
a physical model is the spreading of a liposome on a
mercury electrode [7]: When the liposome interacts with
the mercury surface, it disintegrates and forms an island of
adsorbed lipid molecules. This is accompanied by a change
of double layer capacity, which can be measured as a
current transient. Integration of the current transient gives
the charge transient following Eq. 28 and the form of the
curve shown in Fig. 7.
The origin of time constants is a very complex topic
needing extensive explanations. Here, it may suffice to
mention some possible time depending processes which
can contribute to the measured time constant: (a) diffusion
of particles towards the sensing surface (e.g. in some
electrochemical sensors), (b) convection, when a sensor
chamber (cuvette) has to be filled with solution (e.g. in
optical sensors as used in chromatography), (c) chemical
reactions, esp. in biosensors where enzymatic reactions
may be rather slow. Despite these chemical or physico-
chemical sources of time constants, one should never forget
that each part of a measuring system, from the amplifier to
the recorder, has its own time constant. The modern
instrumentation of chemical analysis normally have so
small constants that they are irrelevant for the measurement
they have been developed for, and the chemical and
physico-chemical sources will dominate; however, it is
good to remember that any measuring system involves time
constants, which occasionally may affect the measurement.
Coming back to the example of spectroscopy (Fig. 4), it
should be mentioned that even modern scanning (!) optical
spectrometers may come to their limits if one tries a too
fast recording.
Chemical reaction kinetics
Chemical reaction kinetics is the study of rates of chemical
processes (reactions). The goal is to find the relations
between the concentrations c of educts or products of a
chemical reaction (as depending variable) and the time t (as
independent variable). In general, all chemical reactions
can be described mathematically by first-order differential
equations. Their solutions, however, depend directly on the
nature of the chemical reaction itself. The latter is char-
acterized by the so-called reaction order, which has
nothing what so ever to do with the order of a differential
equation. The reaction order of a chemical reaction is
simply defined by the sum of exponents of concentrations
occurring in the rate law.
In the following, a first-order chemical reaction, typical
for thermal decompositions or isomerization reactions, is
explained in more detail.
Simple reactions like the transformation of A to B (A ?B) can be described by the differential equation:
�dcA ¼ k � cA � dt ð29Þ
This first-order differential equation describes also a
first-order reaction in chemical kinetics, due to the expo-
nent 1 of the concentration cA. Following a separation of
variables, the integration results in:
ZcAt
cA0
dcA
cA
¼ �k
Z t
0
dt ð30ÞFig. 7 Time response of a sensor which exactly has to be described
by a higher-order differential equation, but which is approximated by
assuming first-order behaviour after a delay time tdelay
1 Page 6 of 12 ChemTexts (2014) 1:1
123
The result
lncAt
cA0
¼ �kt ð31Þ
can be also written in the exponential form:
cAt¼ cA0
e�kt ð32Þ
The half-time, i.e. the time within which the concen-
tration decreases to 50 % of the initial concentration
(Eq. 33), is then given by Eq. 34:
ct1=2¼ 1
2c0 ð33Þ
t1=2 ¼ln 2
k; ð34Þ
which is only dependent on the rate constant k.
In the majority of chemical reactions, however, more
than one educt is involved, e.g.
A þ B! C þ D: ð35Þ
These are second-order reactions in chemical kinetics,
because the sum of exponents of concentrations of A and B
in the rate law (see Eq. 36) is two. In case of a simple
reaction, first-order differential equations are resulting for
the math description:
� dcA
dt¼ k � cA � cB ð36Þ
For equal initial concentrations of A and B (or also for a
dimerization), Reaction 35 can be written as
2A! C þ D, ð37Þ
with the differential equation
� 1
2
dcA
dt¼ k � c2
A ð38Þ
This first-order differential equation is no longer a linear
one, so it will not be considered here. Moreover, in the case
of unequal initial concentrations the solution of the dif-
ferential equation (Eq. 36) has to be developed by expan-
sion into partial fractions. For the latter two situations the
reader is advised to consult textbooks of chemical kinetics
(e.g. [8]).
The radioactive decay
Radioactive decay refers to nuclear conversions of atoms
accompanied by emission of either electromagnetic radia-
tion (gamma radiation) or particles (e.g. alpha particles, i.e.
helium-4 nuclei; beta particles, i.e. electrons or positrons;
protons, fission products, etc.). Although these are physical
processes, they are considered here because the radioactive
decay plays an important role in chemistry, e.g. in isotope
dilution analysis, activation analysis, and in labelling mol-
ecules in kinetic studies, etc. There exists a variety pathways
of nuclear conversions: alpha decay, beta decay, spontane-
ous fission, electron capture, internal conversion, etc., to
name but a few. For a comprehensive overview, see [9, 10]. It
is very interesting to note that among the more than 3,000
isotopes, there are only 265 stable isotopes, all others being
radioactive! Here, we shall treat the simplest case, the decay
of a radioactive isotope which is not produced during the
decay (by another decay or by nuclear activation). The
activity A of a sample containing this isotope is defined as the
number of atoms disintegrating per time unit, i.e. the rate of
decay. The differential expression is:
A ¼ � dN
dtð39Þ
Since each disintegration is completely independent of
any other, and because it is a completely stochastic process,
the rate of disintegration is simply proportional to the
absolute number of radioactive atoms N:
� dN
dt¼ kN ð40Þ
k (Greek letter lambda) is a proportionality constant
called the decay constant. Equation 40 is a first-order dif-
ferential equation which we can also write similar to Eq. 2:
dN
dtþ kN ¼ 0 ð41Þ
Separation of variables leads to:
dN
N¼ �kdt ð42Þ
Integration of Eq. 42 should be done in the limits of N0
(initial number of radioactive atoms) and Nf (final number
of radioactive atoms), when the time runs from 0 to t:
ZNf
N0
dN
N¼ �k
Z t
t¼0
dt ð43Þ
The result is:
ln Nf � ln N0 ¼ �kt ð44Þ
lnNf
N0
¼ �kt ð45Þ
Nf ¼ N0e�kt ð46Þ
Equations 39 and 40 show that generally it holds that
A ¼ kN ð47Þ
This must also hold for N0 and Nf , so that multiplication
of Eq. 46 with k yields:
Af ¼ A0e�kt ð48Þ
ChemTexts (2014) 1:1 Page 7 of 12 1
123
The last equation shows that the activity of a sample
decays in the same way as the number of radioactive atoms.
Equation 46 can be used to calculate the so-called half-
time t1=2 of an isotope: it gives the time in which half of the
initial number of radioactive atoms has decayed:
Nðt1=2Þ ¼N0
2¼ N0e�kt1=2 ð49Þ
This means that
1
2¼ e�kt1=2 ð50Þ
with the result:
t1=2 ¼ln 2
k� 0:693
kð51Þ
If an isotope decays on two different pathways, the overall
decay constant is the sum of the two individual decay con-
stants. Here is an example: 4019K decays on two pathways:
4019K�!kEC 40
18Ar
4019K�!
kb 4020Ca
The indices b (Greek letter beta) and EC indicate a b-
decay and an electron capture decay, respectively.
The differential equation for the overall decay is:
� dNK
dt¼ dNAr
dtþ dNCa
dt¼ kECNK þ kbNK ¼ ðkEC þ kbÞNK
¼ koverallNK
ð52Þ
Relaxation in nuclear magnetic resonance
In an external magnetic field B0, the macroscopic magneti-
zation vector M0 of the spin system is oriented parallel to the
external field, i.e. per definition in z-direction (Mz). Starting
an NMR experiment, this equilibrium state of the spin system
is disturbed: By applying a 90� radio frequency pulse (B1 in
x-direction), the macroscopic magnetization is rotated on the
y-axis or after a 180� pulse the magnetization is turned to �z
(cf. Fig. 8). As a result, the occupation numbers of the spin
energy levels are changed. Switching off this perturbation by
the additional field B1, the spin system returns to its thermal
equilibrium state by relaxation. Felix Bloch [11] described
two different relaxation processes with (i) the spin–lattice or
longitudinal relaxation time T1, describing the relaxation in
the direction of the external magnetic field B0, and (ii) the
spin–spin or transversal relaxation time T2, describing the
relaxation perpendicular to the external magnetic field B0. For
these relaxation processes, first-order mechanisms are
assumed, both from the viewpoint of kinetics (cf. Chapter 4)
and from the viewpoint of mathematics [12]:
dMz
dt¼ �Mz �M0
T1
ð53Þ
dMy
dt¼ �My
T2
ð54Þ
and
dMx
dt¼ �Mx
T2
ð55Þ
These equations represent the time dependence of
magnetization along the three space directions with the rate
constants 1=T1 and 1=T2 for the two relaxation processes
mentioned.
A typical experiment for the determination of the spin–
lattice (or longitudinal) relaxation time T1 is the so-called
inversion recovery experiment. Here, first a 180� pulse
turns the macroscopic magnetization vector M0 from þz to
�z (see Fig. 8). Then, a 90� pulse is necessary to enable the
signal detection in y-direction.
The quantitative description of this experiment starts
from Eq. 53. Introducing a new dependent variable y as
y ¼ Mz �M0 ð56Þ
and
dy ¼ dMz ð57Þ
Equation 53 can be rewritten as
dy
dt¼ � y
T1
ð58Þ
Separation of variables and integration leads to:
Zy
y0
dy
y¼ � 1
T1
Z t
0
dt ð59Þ
Fig. 8 Orientation of the macroscopic magnetization before (þM0)
and after (�M0) a 180� pulse. At t ¼ 0 follows that Mz ¼ �M0
1 Page 8 of 12 ChemTexts (2014) 1:1
123
At t ¼ 0 the definition of y in Eq. 56 gives y0 ¼ �2M0
as initial condition. Integration and re-substitution of y
result in
lnMz �M0
�2M0
� �¼ � t
T1
ð60Þ
or in the exponential notation
M0 �Mz ¼ 2M0e�t=T1 ð61Þ
Mz ¼ M0ð1� 2e�t=T1Þ ð62Þ
Equations 61 and 62 present explicit dependencies of
the magnetization Mz in z-direction on time. This time-
dependent behaviour is schematically illustrated in Fig. 9.
The mathematical solution of the differential equations
(Eq. 54) for the determination of the spin–spin (or trans-
versal) relaxation time T2 is obvious and is exemplarily
given for the time-dependent behaviour of the magnetiza-
tion My in y-direction (cf. Eq. 54). Separation of variables
and integration gives Eqs. 63–65:
ZMy
M0
dMy
My
¼ � 1
T2
Z t
0
dt ð63Þ
lnMy
M0
¼ � t
T2
ð64Þ
My ¼ M0 � e�t=T2 ð65Þ
For t ¼ 0, the magnetization My in y-direction equals M0,
which is the initial value immediately after the 90� pulse.
Equations 64 and 65 can now be used to determine the
spin–spin relaxation time T2, which is a measure how fast
the transversal magnetization disappears. Experimentally,
the well-known spin-echo method developed by Hahn [13]
is used to determine T2.
The RC constant of an electrode
When a metal electrode is in contact with an electrolyte
solution, a double layer is formed at the interface. It has a
complex structure with the charged metal side and the
oppositely charged solution side. The double layer has the
property of a capacitor, as charge can be stored on both
sides, always equal amounts with opposite signs. Of
course, it is also possible that no net charge is on both sides
and this situation is referred to as the potential of zero
charge (given versus a reference potential). Most electro-
chemical techniques make use of a changing electrode
potential (linear and non-linear) to measure the current
response. Only rarely the current is deliberately changed or
controlled to measure the response of the electrode
potential. In the potential-controlled experiments, the
double layer is always charged or discharged during the
experiment, depending on the applied potentials, so that
charging currents accompany the so-called faradaic cur-
rents, which result from charge transfer reactions at the
interphase. Thus, the charging or capacitive currents are
normally interfering as they may become dominating when
the compounds undergo the electrochemical reaction are at
very low concentration. Therefore, the double layer
charging is worth to be considered, not to speak about their
great importance for modern capacitors. It is also important
to study capacitive currents because the double layer
charging reveals information about the complex structure
of the solution and metal sides of the interface. Here, we
shall discuss the most simple case: two metal pieces are
inserted in an electrolyte solution, e.g. of potassium nitrate
in water, and there is no compound in the solution other
than KNO3 and H2O, i.e. only the following ions and
molecules are present: K?, NO�3 ; H2O, H3Oþ; OH� (if
we neglect more complex ions like H5Oþ2 , and also ion
pairs, like [KNO3], which are present only in extremely
small concentrations). If at each electrode in the above
system the potential difference between the metal and the
solution side is insufficient to perform an electrochemical
reaction with all these species, i.e. insufficient to reduce the
protons of water to hydrogen or oxidize the oxygen-con-
taining species to oxygen, any change of the voltage
between the electrodes can only charge or discharge the
double layer of the electrodes. In electrochemistry, such
electrode is called an ideally polarizable electrode. The
current has thus to flow via a resistor, which is formed by
the electrolyte solution into or out of the capacitor formed
by the double layers. All other resistors in the circuit, e.g.Fig. 9 Development of the macroscopic magnetization in z-direction
Mz in dependence on time following a 180� pulse
ChemTexts (2014) 1:1 Page 9 of 12 1
123
the metal wires, have a much lower resistance than the
electrolyte solution. In such case, we may model the situ-
ation at one electrode by the capacitor ‘‘double layer’’, and
the resistor ‘‘solution’’ in series (see Fig. 10). (The second
electrode can be treated in the same way). Let us suppose
that the double layer is initially completely discharged (the
electrode rests at the potential of zero charge), i.e. there is
no charge separated by the two capacitor plates. In a
potential step experiment (i.e. when the electrode potential
is stepped from one value to another), the double layer is
charged (the capacitor is loaded). At the beginning (t ¼ 0),
the current flows only through the resistor, and upon
charging the current decays to zero, when the capacitor is
charged to its potential difference DE.
According to Kirchhoff’s law, the overall potential dif-
ference across the resistor and capacitor is the sum of two
potential drops:
DE ¼ DEresistor þ DEcapacitor ð66Þ
For the potential drop across the resistor follows (Ohm’s
law):
DEresistor ¼ RI ð67Þ
where R is the solution resistance and I the current.
The potential drop across the capacitor is:
DEcapacitor ¼q
Cð68Þ
where q is the charge, and C the capacitance of the
capacitor ‘‘double layer’’ (which in fact is depending on the
electrode potential, what we may neglect here).
Substituting the terms from Eqs. 67 and 68 in Eq. 66
yields:
DE � RI � q
C¼ 0 ð69Þ
Solving this equation for I gives:
I ¼ DE
R� q
RCð70Þ
Since the current is the first derivative of charge over
time (I ¼ dq=dt), one can write Eq. 70 also as follows:
dq
dt¼ DE
R� q
RC¼ � 1
RCðq� C � DEÞ ð71Þ
dq
dtþ 1
RCðq� C � DEÞ ¼ 0 ð72Þ
Equation 72 is a nonhomogeneous linear first-order
differential equation. Integration of Eq. 72 in the limits of 0
and q, and 0 and t gives:
Zq
0
dq
q� C � DE¼ � 1
RC
Z t
0
dt ¼ � t
RCð73Þ
At t ¼ 0 follows q ¼ 0 from the assumption that the
electrode is initially at the potential of zero charge. To
solve the integral on the left side, one needs to make the
following substitution:
q0 ¼ q� C � DE ð74Þ
from which follows q ¼ q0 þ C � DE and dq ¼ dq0.
Zq0t
q0t¼0
dq0
q0¼ � t
RCð75Þ
(This is equal to solving the differential equationdq0
dtþ 1
RCq0 ¼ 0).
½ln q0�q0t
q0t¼0
¼ � t
RCð76Þ
Back-substitution q0 ¼ q� C � DE and dq0 ¼ dq and
defining q0t¼0 ¼ �C � DE (this equals to the condition
qt¼0 ¼ 0) lead to:
lnðq� C � DEÞ � lnð�C � DEÞ ¼ � t
RCð77Þ
lnq� C � DE
�C � DE
� �¼ � t
RCð78Þ
� q� C � DE
C � DE
� �¼ e�t=RC ð79Þ
q� C � DE ¼ �C � DE � e�t=RC ð80Þ
q ¼ �C � DE � e�t=RC þ C � DE ð81Þ
q ¼ C � DEð1� e�t=RCÞ ð82Þ
Since I ¼ dq=dt it follows from Eq. 82 that:
I ¼ dq
dt¼ � 1
RCð�C � DEÞ � e�t=RC ¼ DE
R� e�t=RC ð83Þ
Since the DE=R must be equal to the initial current I0 at
t ¼ 0, Eq. 83 gives the current transient:
I ¼ I0 � e�t=RC ð84Þ
describing the exponential decay of current following a
potential step. RC is also called the time constant s and one
can write:
I ¼ I0 � e�t=s ð85ÞFig. 10 The resistor formed by the electrolyte solution having the
resistance R and the capacitor formed by the double layer of the
electrode having the capacitance C in an electrochemical cell
1 Page 10 of 12 ChemTexts (2014) 1:1
123
Another possibility to solve the nonhomogeneous linear
first-order differential Eq. 72, consists in solving first the
homogeneous equation
dqh
dtþ 1
RCqhðtÞ ¼ 0 ð86Þ
yielding qhðtÞ as usual in an exponential form. After that, a
particular solution qpðtÞ of the nonhomogeneous differen-
tial equation can be found applying the equation
qpðtÞ ¼ uðtÞ � qhðtÞ ð87Þ
and its first derivative
q0pðtÞ ¼ u0ðtÞ � qhðtÞ þ uðtÞ � q0hðtÞ ð88Þ
Both expressions for qpðtÞ and for its first derivative
q0pðtÞ are substituted in Eq. 72, which finally leads to the
determination of uðtÞ and thus to qpðtÞ. The complete
solution of Eq. 72 is then the sum of the solution of the
homogeneous differential equation qhðtÞ and the particular
solution qpðtÞ, i.e.
qðtÞ ¼ qhðtÞ þ qpðtÞ: ð89Þ
Both mathematical approaches result in Eq. 85.
In many experiments, it is crucial to know the value of
the RC time constant of the electrode/electrolyte system,
because the charging current may be superimposed to
faradaic currents that are investigated.
For a more detailed understanding of the used electro-
chemical terms, we suggest to consult an electrochemical
dictionary [14].
Conclusions
Table 1 gives an overview of the discussed cases of
application of first-order differential equations to
chemistry.
It should be mentioned that there are many other pro-
cesses in science, which are based on first-order differential
equations, e.g. the so-called ‘‘exponential growth’’ of
bacteria cultures, the barometric formula or Newton’s law
of cooling.
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Table 1 Overview of the discussed cases of applications of a first-order differential equation a1dy
dxþ a2y ¼ 0
Differential equation a1 a2 y x Integrated equation
Bouguer–Lambert–
Beer law� dIðxÞ
dx� jcIðxÞ ¼ 0 -1 �jc I x ln I0
I¼ jcxe
Time constant of a
sensora1
dzdtþ a2zðtÞ ¼ 0 for the equation:
S ¼ Smax � zðtÞa1 a2 z t zðtÞ ¼ z0e
�a2a1
tyielding: S ¼
Smax 1� e�ts
� �with a1
a2¼ s
Kinetics of chemical
reaction
dcA
dtþ kcA ¼ 0 þ1 þk cA t cAt
¼ cA0e�kt
Radioactive decay dNdtþ kN ¼ 0 1 k N t Nf ¼ N0e�kt
Bloch equation (a) dMz
dtþ Mz�M0
T1¼ 0 þ1 1=T1 Mz �M0 t Mz ¼ M0 1� 2e�t=T1
� �Bloch equation (b) dMy
dtþ My
T2¼ o þ1 1=T2 My t My ¼ M0 � e�t=T2
Resistor and capacitor
in series
dq0
dtþ 1
RCq0 ¼ 0 1 1
RC q0
q0 ¼ q� C � DE
tln q0½ �q
0t
q0t¼0¼ � t
RC
q ¼ C � DE 1� e�t=RC� �
I ¼ DE
R� e�t=RC ¼ I0 � e�t=RC
ChemTexts (2014) 1:1 Page 11 of 12 1
123
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